Lemma 8.5.15. Let \(R \in \CAlg (\Sp _{\geq 0})\) be a connective commutative ring spectrum, and let \(M\) and \(N\) be flat \(R\)-modules. Then \(M \otimes _R N\) is again flat.
Proof. By Corollary 8.2.5, connectivity and coconnectivity of an \(R\)-module are detected on its underlying spectrum. A connective \(R\)-module \(P\) is therefore flat if and only if the functor \((-) \otimes _R P\colon \Mod _R \to \Mod _R\) is t-exact, as opposed to merely its composite with the forgetful functor to \(\Sp \). Now associativity of the relative tensor product provides a natural isomorphism \[ (-) \otimes _R (M \otimes _R N) \quad \simeq \quad \bigl ((-) \otimes _R M\bigr ) \otimes _R N \] of functors \(\Mod _R \to \Mod _R\). Since \(M\) and \(N\) are flat, the right-hand side is a composite of two t-exact functors, hence t-exact. Finally, \(M \otimes _R N\) is connective by Observation 8.5.9, so it is flat. □
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